A class of perfect fluids in general relativity

This thesis is concerned with exact solutions of Einstein's field equations of general relativity, in particular, when the source of the gravitational field is a perfect fluid with a purely electric Weyl tensor. General relativity, cosmology and computer algebra are discussed briefly. A mathematical introduction to Riemannian geometry and the tetrad formalism is then given. This is followed by a review of some previous results and known solutions concerning purely electric perfect fluids. In addition, some orthonormal and null tetrad equations of the Ricci and Bianchi identities are displayed in a form suitable for investigating these space-times. Conformally flat perfect fluids are characterised by the vanishing of the Weyl tensor and form a sub-class of the purely electric fields in which all solutions are known (Stephani 1967). The number of Killing vectors in these space-times is investigated and results presented for the non-expanding space-times. The existence of stationary fields that may also admit 0, 1 or 3 spacelike Killing vectors is demonstrated. Shear-free fluids in the class under consideration are shown to be either non-expanding or irrotational (Collins 1984) using both orthonormal and null tetrads. A discrepancy between Collins (1984) and Wolf (1986) is resolved by explicitly solving the field equations to prove that the only purely electric, shear-free, geodesic but rotating perfect fluid is the Godel (1949) solution. The irrotational fluids with shear are then studied and solutions due to Szafron (1977) and Allnutt (1982) are characterised. The metric is simplified in several cases where new solutions may be found. The geodesic space-times in this class and all Bianchi type 1 perfect fluid metrics are shown to have a metric expressible in a diagonal form. The position of spherically symmetric and Bianchi type 1 space-times in relation to the general case is also illustrated.

Squaring the circle: the cultural relativity of 'good' shape

The Gestalt theorists of the early twentieth century proposed a psychological primacy for circles, squares and triangles over other shapes. They described them as 'good' shapes and the Gestalt premise has been widely accepted. Rosch (1973), for example, suggested that shape categories formed around these 'natural' prototypes irrespective of the paucity of shape terms in a language. Rosch found that speakers of a language lacking terms for any geometric shape nevertheless learnt paired-associates to these 'good' shapes more easily than to asymmetric variants. We question these empirical data in the light of the accumulation of recent evidence in other perceptual domains that language affects categorization. A cross-cultural investigation sought to replicate Rosch's findings with the Himba of Northern Namibia who also have no terms in their language for the supposedly basic shapes of circle, square and triangle. A replication of Rosch (1973) found no advantage for these 'good' shapes in the organization of categories. It was concluded that there is no necessary salience for circles, squares and triangles. Indeed, we argue for the opposite because these shapes are rare in nature. The general absence of straight lines and symmetry in the perceptual environment should rather make circles, squares and triangles unusual and, therefore, less likely to be used as prototypes in categorization tasks. We place shape as one of the types of perceptual input (in philosophical terms, 'vague') that is readily susceptible to effects of language variation.